Conti is a theatre, film and television actor. He began working with the Dundee Repertory in 1959. He appeared on Broadway in Whose Life Is It Anyway? in 1979, and in London he played the lead in Jeffrey Bernard is Unwell at the Garrick Theatre.
Besides taking the leading role in the TV versions of Frederic Raphael's The Glittering Prizes and Alan Ayckbourn's The Norman Conquests, Conti appeared in the "Princess and the Pea" episode of the family television series Faerie Tale Theatre, guest-starred on Friends and Cosby, and played opposite Nigel Hawthorne in a long-running series of Vauxhall Astra car advertisements in the United Kingdom during the mid-1990s.
Conti's novel The Doctor, about a former secret operations pilot for intelligence services, was published in 2004.
Conti considered running as the Conservative candidate in the 2008 London mayoral election, but did not, and in the following election in 2012 he supported unsuccessful independent candidate Siobhan Benita. In the run up to the 2015 general election, Conti said in an interview published in several newspapers that he had come to view socialism as a religion with a "vicious, hostile spirit" and that "conservatism was about enabling people to improve their lives."
Conti participated in a genetic-mapping project conducted by the company ScotlandsDNA (now called BritainsDNA). In 2012 Conti and the company announced that Conti shares a genetic marker with Napoléon Bonaparte. Conti has said that he "burst out laughing" when told he was directly related to Napoléon on his father's side.
Conti is a prominent resident of Hampstead in North West London, having lived in the area for several decades. Conti was part of a campaign against the opening of a Tesco supermarket in nearby Belsize Park. Conti put his Hampstead house up for sale in 2015 for £17.5 million after his long-running opposition to the building plans of his neighbour, the footballer Thierry Henry. Conti had also opposed development plans for Hampstead's Grove Lodge, the 18th-century Grade II listed former home of Novelist John Galsworthy.